Countless papers in fine-grained complexity and graph algorithms study triangles and short cycles.
Let us mention a few that are more relevant for this work.
Roditty and Vassilevska Williams [1]} proved a conditional lower bound for a particular approach for approximating the girth of a graph, but left proving hardness (for any algorithm) as a main open question.
Dahlgaard, Knudsen, and Stöckel [2]} prove the hardness of \(k\) -cycle detection for all \(k\ge 5\)  assuming the hardness of the \(k=3\)  and \(k=4\)  cases.
Dudek and Gawrychowski prove that counting 4-cycles is equivalent to computing the quartet distance on trees [3]} and to counting 4-patterns in permutations [4]}.
Unlike in undirected graphs where the \(k\) -Cycle detection problem tends to become easier as \(k\)  grows (because the graph gets sparser), this is not the case in directed graphs where it is conjectured that \(n^{2-o(1)}\)  is required for large enough \(k\ge 3\) , and this conjecture is implied by other conjectures on the \(k\) -Clique problem [5]}, [6]}.
